Section:
New Results
Dimension free principal component analysis
Participants :
Olivier Catoni, Ilaria Giulini.
In a work in progress, Ilaria Giulini, as part of her PhD studies,
proved the following dimension free inequality, related to
Principal Component Analysis in high dimension. Given an i.i.d.
sample , of vector valued random variables
, there exists an estimator of the
quadratic form
such that for any , with probability at least ,
for any ,
where
where is the Gram matrix and where
is some kurtosis coefficient.
This result proves that the expected energy in direction
can be estimated at a rate that is independent of the dimension
of the ambient space . It is obtained using
PAC-Bayes inequalities with Gaussian parameter perturbations.
The same bound holds in
a Hilbert space of infinite dimension, opening the possibility of
a rigorous mathematical study of kernel principal component analysis
of random data,
where the data are represented in a possibly infinite dimensional
reproducing kernel Hilbert space.